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A dynamic epistemic characterization of backward induction without counterfactuals

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  • Giacomo Bonanno

    (Department of Economics, University of California Davis)

Abstract

The analysis of rational play in dynamic games is usually done within a static framework that specifies a player's initial beliefs as well as his disposition to revise those beliefs conditional on hypothetical states of information. We suggest a simpler approach, where the rationality of a player's choice is judged on the basis of the actual beliefs that the player has at the time he has to make that choice. We propose a dynamic framework where the set of "possible worlds" is given by state-instant pairs (w,t). Each state w specifies the entire play of the game and, for every instant t, (w,t) specifies the history that is reached at that instant (in state w). A player is said to be active at (w,t) if the history reached in state w at date t is a decision history of his. At every state-instant pair (w,t) the beliefs of the active player provide an answer to the question "what will happen if I take action a", for every available action a. A player is said to be rational at (w,t) if either he is not active there or the action he ends up taking at state w is "optimal" given his beliefs at (w,t). We provide a characterization of backward induction in terms of the following event: the first mover (i) is rational and has correct beliefs, (ii) believes that the active player at date 1 is rational and has correct beliefs, (iii) believes that the active player at date 1 believes that the active player at date 2 is rational and has correct beliefs, etc. Thus our epistemic characterization does not rely on dispositional belief revision or on (objective or subjective) counterfactuals.

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Bibliographic Info

Paper provided by University of California, Davis, Department of Economics in its series Working Papers with number 122.

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Length: 17
Date of creation: 18 Mar 2012
Date of revision:
Handle: RePEc:cda:wpaper:12-2

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Related research

Keywords: Perfect-information game; backward induction; dynamic interactive beliefs; rationality; Kripke frame;

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References

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  1. Perea,Andrés, 2012. "Epistemic Game Theory," Cambridge Books, Cambridge University Press, number 9781107008915, April.
  2. Samet, Dov, 1996. "Hypothetical Knowledge and Games with Perfect Information," Games and Economic Behavior, Elsevier, vol. 17(2), pages 230-251, December.
  3. Battigalli, Pierpaolo & Siniscalchi, Marciano, 2002. "Strong Belief and Forward Induction Reasoning," Journal of Economic Theory, Elsevier, vol. 106(2), pages 356-391, October.
  4. Battigalli, Pierpaolo & Bonanno, Giacomo, 1999. "Recent results on belief, knowledge and the epistemic foundations of game theory," Research in Economics, Elsevier, vol. 53(2), pages 149-225, June.
  5. Halpern, Joseph Y., 2001. "Substantive Rationality and Backward Induction," Games and Economic Behavior, Elsevier, vol. 37(2), pages 425-435, November.
  6. Perea,Andrés, 2012. "Epistemic Game Theory," Cambridge Books, Cambridge University Press, number 9781107401396, April.
  7. Balkenborg, Dieter & Eyal Winter, 1995. "A Necessary and Sufficient Epistemic Condition for Playing Backward Induction," Discussion Paper Serie B 331, University of Bonn, Germany.
  8. Stalnaker, Robert, 1996. "Knowledge, Belief and Counterfactual Reasoning in Games," Economics and Philosophy, Cambridge University Press, vol. 12(02), pages 133-163, October.
  9. Pierpaolo Battigalli & Alfredo Di Tillio & Dov Samet, 2011. "Strategies and interactive beliefs in dynamic games," Working Papers 375, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
  10. Ben-Porath, Elchanan, 1997. "Rationality, Nash Equilibrium and Backwards Induction in Perfect-Information Games," Review of Economic Studies, Wiley Blackwell, vol. 64(1), pages 23-46, January.
  11. Aumann, Robert J., 1995. "Backward induction and common knowledge of rationality," Games and Economic Behavior, Elsevier, vol. 8(1), pages 6-19.
  12. Adam Brandenburger, 2007. "The power of paradox: some recent developments in interactive epistemology," International Journal of Game Theory, Springer, vol. 35(4), pages 465-492, April.
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Cited by:
  1. Zuazo Gain, Peio, 2014. "Uncertain Information Structures and Backward Induction," IKERLANAK Ikerlanak;2014-79, Universidad del País Vasco - Departamento de Fundamentos del Análisis Económico I.

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